Graded ring

In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading.

A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded Z-algebra.

The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to a non-associative algebra as well; e.g., one can consider a graded Lie algebra.

First properties


be a graded ring. Elements of any factor of the decomposition are called homogeneous elements of degree n. Every element a of A may be uniquely written as a sum a = a1 + a2 + ... + an with all ai homogeneous elements of distinct Ai. These ai are called the homogeneous components of a.

Some basic properties are:

An ideal is homogeneous if, for every element , its homogeneous components belong also to (Equivalently, they are graded submodules of A; see #Graded module.) The intersections of a homogeneous ideal with the are called the homogeneous parts of . A homogeneous ideal is the direct sum of its homogeneous parts.

If I is a homogeneous ideal in A, then is also a graded ring, and has decomposition

Basic examples

Graded module

The corresponding idea in module theory is that of a graded module, namely a left module M over a graded ring A such that also


Example: a graded vector space is an example of a graded module over a field (with the field having trivial grading.)

Example: a graded ring is a graded module over itself. An ideal in a graded ring is homogeneous if and only if it is a graded submodule. The annihilator of a graded module is a homogeneous ideal.

Example: Given an ideal I in a commutative ring R and an R-module M, is a graded module over the associated graded ring .

A morphism between graded modules, called a graded morphism, is a morphism of underlying modules that respects grading; i.e., . A graded submodule is a submodule that is a graded module in own right and such that the set-theoretic inclusion is a morphism of graded modules. Explicitly, a graded module N is a graded submodule of M if and only if it is a submodule of M and satisfies . The kernel and the image of a morphism of graded modules are graded submodules.

Remark: To give a graded morphism from a graded ring to a graded ring with the image lying in the center is the same as to give the structure of a graded algebra to the latter ring.

Given a graded module M, the -twist of is a graded module defined by . (cf. Serre's twisting sheaf in algebraic geometry.)

Let M and N be graded modules. If is a morphism of modules, then f is said to have degree d if . An exterior derivative of differential forms in differential geometry is an example of such a morphism having negative degree.

Invariants of graded modules

Given a graded module M over a commutative graded ring A, one can associate the formal power series :

(assuming are finite.) It is called the Hilbert–Poincaré series of M.

A graded module is said to be finitely generated if the underlying module is finitely generated. The generators may be taken to be homogeneous (by replacing the generators by their homogeneous parts.)

Suppose A is a polynomial ring , k a field, and M a finitely generated graded module over it. Then the function is called the Hilbert function of M. The function coincides with the integer-valued polynomial for large n called the Hilbert polynomial of M.

Graded algebra

An algebra A over a ring R is a graded algebra if it is graded as a ring.

In the usual case where the ring R is not graded (in particular if R is a field), it is given the trivial grading (every element of R is of grade 0). Thus, RA0 and the Ai are R-modules.

In the case where the ring R is also a graded ring, then one requires that


In other words, we require A to be a left and right graded module over R.

Examples of graded algebras are common in mathematics:

Graded algebras are much used in commutative algebra and algebraic geometry, homological algebra and algebraic topology. One example is the close relationship between homogeneous polynomials and projective varieties. (cf. homogeneous coordinate ring.)

G-graded rings and algebras

The above definitions have been generalized to gradings ring using any monoid G as an index set. A G-graded ring A is a ring with a direct sum decomposition

such that

The notion of "graded ring" now becomes the same thing as a N-graded ring, where N is the monoid of non-negative integers under addition. The definitions for graded modules and algebras can also be extended this way replacing the indexing set N with any monoid G.




Some graded rings (or algebras) are endowed with an anticommutative structure. This notion requires a homomorphism of the monoid of the gradation into the additive monoid of Z/2Z, the field with two elements. Specifically, a signed monoid consists of a pair (Γ, ε) where Γ is a monoid and ε : Γ → Z/2Z is a homomorphism of additive monoids. An anticommutative Γ-graded ring is a ring A graded with respect to Γ such that:

xy=(-1)ε (deg x) ε (deg y)yx, for all homogeneous elements x and y.


See also


  1. Matsumura 1986, Theorem 13.1
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